A measure preserving system (or just “system") \((X,\mu ,T)\)  is a probability measure space \((X,\mu )\) , together with a map \(T:X\rightarrow X\)  preserving \(\mu \) : \(\mu (T^{-1}D) = \mu (D)\)  for every measurable \(D\subseteq X\) . In this article, \(T\)  will always be invertible. We do not mention the \(\sigma \) -algebra of measurable sets on \(X\) , as we only ever consider measurable subsets of \(X\) . See any of [1]}, [2]}, [3]}, [4]}, [5]}, [6]} for standard definitions regarding measure preserving systems, including ergodicity, weak mixing, and the definition of isomorphism for measure preserving systems.
